ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-11

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The Cantor function is continuous and of bounded variation but not absolutely continuous

Example

The Cantor function $c : [0, 1] \to [0, 1]$ is continuous and nondecreasing, so it has total variation one, but it is not absolutely continuous.

Facts & Assumptions

Given: The Cantor function and its standard stage construction.

[L1]

The Cantor function is continuous and nondecreasing, with $c (0) = 0$ and $c (1) = 1$ (The Cantor function is continuous on $[0, 1]$ , The Cantor function is well defined, satisfies $c (x) \leq c (y)$ whenever $x \leq y$ , is surjective onto $[0, 1]$ , and is constant on every interval removed from the Cantor set).

[L2]

At stage $n$ , the $2^{n}$ surviving closed intervals have total length $(2 / 3)^{n}$ , and $c$ increases by $2^{- n}$ across each.

[L3]

The sequence $(2 / 3)^{n}$ tends to zero (For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ ).

Verification

technique · direct

1.1

Monotonicity makes every variation sum telescope after its absolute values are removed, so ${Var}_{[0, 1]} (c) = c (1) - c (0) = 1$ . Thus $c$ is BV.

2.1

For the finite disjoint family of the $2^{n}$ surviving intervals, the total length is $(2 / 3)^{n}$ but the sum of endpoint increments is $2^{n} 2^{- n} = 1$ . By [L3], the former is eventually smaller than every prescribed $δ > 0$ , while the latter never falls below, say, $1 / 2$ . This contradicts the definition of absolute continuity.

L2L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

60 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Christopher Heil, Absolute Continuity and the Banach-Zaretsky Theorem, Example 3.3 (standard reference, not scraped)